Incentives to reduce crop trait durability.

The equilibria in the noncommitment case are identical to ones we obtained in the commitment case. Not surprisingly, the findings obtained with commitment are still valid in the noncommitment case: a monopolist who cannot commit on future prices adopts a durable good strategy. She sells to all of the farmers in the first period and to none of them in the second period. The introduction of a self-production fee increases efficiency by reducing self-production.

Inbred Line Monopoly Pricing and Discrimination

In the previous analysis, implicitly, no price discrimination among farmers was allowed. Indeed, we have not considered the case where the monopolist can sell inbred line seed in the second period at different prices depending on whether farmers bought seed in the first period. (18)

It is easy to show that by discriminating, the monopolist can extract all of the surplus if she commits on future prices. Indeed, by pricing at 2[[PI].sub.L] in the first period, with the promise of providing free seed in the second period, she sells to every farmer at the farmers' surplus 2[[PI].sub.L]. However, in the second period, she has an incentive not to keep her promise and to sell the seed at a positive price. Expecting this behavior, no farmers (with strictly positive self-production costs) buy at 2[[PI].sub.L]. Therefore, we investigate monopoly pricing with discrimination but without commitment on future prices. In addition to the per period prices [P.sub.1L] and [P.sub.2L], we introduce a special second-period price [[??].sub.2L] for farmers who purchased seed in the first period. We solve by backwards induction.

In the second period, farmers who did not buy the seed in the first period represent captive demand and, thus, the monopolist can set the price [P.sub.2L] = [[PI].sub.L]. Those who did buy seed in the first period are those with a low [theta]. Indeed, if it is optimal for a farmer [theta]' to buy seed in the first period, it is optimal for every farmer [theta] < [theta]'. We denote [??] as the farmer who is indifferent between buying during both periods and buying only in the second period. Among farmers who buy in the first period, those with the highest self production cost ([theta] > [[??].sub.2L]) prefer to buy seed in the second period. In the second period, the monopolist maximizes [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with respect to [[??].sub.2L]. The first-order condition yields [[??].sub.2L] = [??]/2.

Among farmers who bought seed in the first period, those with self-production cost [theta] higher than [[??].sub.2L], also buy seed in the second period. Therefore, farmers with self-production costs [theta] [member of] [[??]/2, [??]] buy seed in each period. They obtain the same surplus 2[[PI].sub.L] - [P.sub.1L] - [??]/2. This surplus is nil because, by definition, the farmer [??] is indifferent between buying during both periods and buying only in the second period, in which case he gets [[PI].sub.L] - [P.sub.2L] = 0. Hence, [P.sub.1L] = 2[[PI].sub.L] - [??]/2. We assume that in the case of indifference, farmers prefer to buy in both periods, which implies that all of the farmers who do not self-produce buy seed in each period. Therefore, [??] = [theta], which implies [[??].sub.2L] = [bar.[theta]]/2 and [P.sub.1L] - 2[[PI].sub.L] - [bar.[theta]]/2.

A discriminating monopolist sells seed to all of the farmers in the first period at a higher price than without discrimination (cf., table 2, durable good strategy). All of the farmers buy, either because they have low self-production costs (and then self-produce) or expect to buy at price [[??].sub.2L] = [bar.[theta]]/2, which is lower than [P.sub.2L] = [[PI].sub.L]. Half of the farmers self-produce at a cost [theta] [less than or equal to] [bar.[theta]]/2, and thus obtain a positive surplus [bar.[theta]]/2 - [theta]. The other half, with high self-production costs, buy seed in each period and obtain zero surplus. Hence, price discrimination leads to a reduction of self-production. The monopolist extracts all of the surplus from the latter farmers and only 2[[PI].sub.L] - [bar.[theta]]/2 from those who self-produce. She thus obtains strictly more from all of the farmers by discriminating.

The introduction of a self-production fee raises the farmer's self-production outside option. It allows the monopolist to increase the second-period price to ([bar.[theta]] + [tau])/2. This reduces self-production from farmers with [theta] [less than or equal to] ([bar.[theta]] - [tau])/2. The first-period price decreases at 2[[PI].sub.L] - ([bar.[theta]] + [tau])/2. The royalty fee increases the monopoly payoff. Indeed, in addition to reducing self-production, it also increases the surplus extracted from the farmers who self-produce to 2[[PI].sub.L]- [bar.[theta]]/2 + [tau]/2. Since fewer farmers self-produce, it also increases efficiency.

By setting different second-period prices for farmers who self-produce and those who do not, a discriminating monopolist increases her payoff by reducing self-production. The introduction of a fee reduces self-production even more, and increases efficiency.

Multiseed Production

We now analyze what happens when the monopolist can sell both types of seeds, in the case of commitment on future prices when there is no price discrimination. (19)

If hybrid seed is more efficient than inbred line seed, i.e., c < [DELTA][PI], the monopolist prefers to sell only hybrid seed, because she then extracts all of the surplus, and this surplus is greater with hybrid seed than inbred.

However, if inbred line seed is more efficient than hybrid seed, i.e., c > [DELTA][PI], the monopolist can introduce hybrid seed for discriminatory purposes. Recall that when selling only inbred line seed, the monopolist is constrained to set a reasonable price in the first period, as otherwise some farmers would prefer not to buy, and this would reduce the monopoly payoff. The constraint is relaxed in the multiseed case because, by selling hybrid seed in the first period, the monopolist can earn some payoff from farmers who do not buy inbred line seed.

For the monopolist, the optimal second-period inbred line seed price is [[PI].sub.L] and the optimal first-period hybrid seed price is [[PI].sub.H]. With such prices, any farmer can at least earn no surplus by buying inbred line seed in the first period and hybrid seed in the second period. (20) The alternative for farmers is to buy inbred line seed in the first period and self-produce. Their surplus is then 2[[PI].sub.L] - [P.sub.1L] - [theta]. Hence, all of the farmers with [theta] < 2[[PI].sub.L] - [P.sub.1L] prefer to buy inbred line seed in the first period, and the rest prefer to buy hybrid seed. The monopoly's program is (21)

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We do not describe the details of the solution and we only compare this multiseed strategy to the mono-seed strategy analyzed before. Recall that when only inbred seed is sold, the monopolist adopts a durable good strategy, and then all of the farmers buy seed in the first period and self-produce.

We show that the monopoly payoff is higher with a multiseed strategy only if c [[DELTA][PI], [DELTA][PI] + 2[bar.[theta]]]. The adoption of the multiseed strategy also affects the farmers' surplus, as some farmers switch from buying inbred line seed in the first period and self-producing, to buying hybrid seed in the first period and inbred line seed in the second period. This switch has an ambiguous effect on the surplus. On one hand, there is a loss in the first period because these farmers use a less efficient hybrid seed. On the other hand, there is a gain in the second period because the inbred line seed they use is produced at a lower cost (0 instead of [theta]). In fact, we show that the multiseed strategy increases the surplus if c [less than or equal to] [DELTA][PI] + 2[bar.[theta]]/3.

In summary, when the monopolist has the option to sell both hybrid and inbred line seeds during the same period, she produces both technologically dominated hybrid seed and inbred line seed if c [member of] [[DELTA][PI], [DELTA][PI] + 2[bar.[theta]]]. This is efficient only when c [less than or equal to] [DELTA][PI] + 2[bar.[theta]]/3.

A fee on self-produced seed has a similar impact on the incentive to introduce hybrid seed as does the mono-seed monopolist case examined before. In a nutshell, as in the monoseed case, a low fee has no impact on this decision. However, beyond a threshold, the fee reduces the range of the inefficient introduction of technologically dominated hybrid seed. This range shrinks as the fee increases.

Therefore, a multiseed monopolist produces both seeds even though hybrid seed is technologically dominated. The introduction of a self-production fee reduces the inefficient introduction of hybrid seed.